Cubic critical, wide range

Percentage Accurate: 17.9% → 99.9%

Time: 14.2s

Alternatives: 4

Speedup: 2.9×

Specification

\[\left(\left(4.930380657631324 \cdot 10^{-32} < a \land a < 2.028240960365167 \cdot 10^{+31}\right) \land \left(4.930380657631324 \cdot 10^{-32} < b \land b < 2.028240960365167 \cdot 10^{+31}\right)\right) \land \left(4.930380657631324 \cdot 10^{-32} < c \land c < 2.028240960365167 \cdot 10^{+31}\right)\]

Math

\[\begin{array}{l} \\ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))

C

double code(double a, double b, double c) {
	return (-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a);
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (-b + sqrt(((b * b) - ((3.0d0 * a) * c)))) / (3.0d0 * a)
end function

Java

public static double code(double a, double b, double c) {
	return (-b + Math.sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a);
}

Python

def code(a, b, c):
	return (-b + math.sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c)))) / Float64(3.0 * a))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a);
end

Wolfram

code[a_, b_, c_] := N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision]

Initial Program: 17.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))

C

double code(double a, double b, double c) {
	return (-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a);
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (-b + sqrt(((b * b) - ((3.0d0 * a) * c)))) / (3.0d0 * a)
end function

Java

public static double code(double a, double b, double c) {
	return (-b + Math.sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a);
}

Python

def code(a, b, c):
	return (-b + math.sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c)))) / Float64(3.0 * a))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a);
end

Wolfram

code[a_, b_, c_] := N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \frac{c}{\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)}} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ c (- (- b) (sqrt (fma b b (* c (* a -3.0)))))))

C

double code(double a, double b, double c) {
	return c / (-b - sqrt(fma(b, b, (c * (a * -3.0)))));
}

Julia

function code(a, b, c)
	return Float64(c / Float64(Float64(-b) - sqrt(fma(b, b, Float64(c * Float64(a * -3.0))))))
end

Wolfram

code[a_, b_, c_] := N[(c / N[((-b) - N[Sqrt[N[(b * b + N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 19.0%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
2. Add Preprocessing

4. Applied egg-rr 19.0%

   \[\leadsto \color{blue}{\frac{-1}{\frac{3 \cdot a}{b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}}}} \]
5. Step-by-step derivation

   1. associate-/r/ N/A

      \[\leadsto \color{blue}{\frac{-1}{3 \cdot a} \cdot \left(b - \sqrt{a \cdot \left(-3 \cdot c\right) + b \cdot b}\right)} \]

   2. flip-- N/A

      \[\leadsto \frac{-1}{3 \cdot a} \cdot \color{blue}{\frac{b \cdot b - \sqrt{a \cdot \left(-3 \cdot c\right) + b \cdot b} \cdot \sqrt{a \cdot \left(-3 \cdot c\right) + b \cdot b}}{b + \sqrt{a \cdot \left(-3 \cdot c\right) + b \cdot b}}} \]

   3. associate-*r/ N/A

      \[\leadsto \color{blue}{\frac{\frac{-1}{3 \cdot a} \cdot \left(b \cdot b - \sqrt{a \cdot \left(-3 \cdot c\right) + b \cdot b} \cdot \sqrt{a \cdot \left(-3 \cdot c\right) + b \cdot b}\right)}{b + \sqrt{a \cdot \left(-3 \cdot c\right) + b \cdot b}}} \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{-1}{3 \cdot a} \cdot \left(b \cdot b - \sqrt{a \cdot \left(-3 \cdot c\right) + b \cdot b} \cdot \sqrt{a \cdot \left(-3 \cdot c\right) + b \cdot b}\right)}{b + \sqrt{a \cdot \left(-3 \cdot c\right) + b \cdot b}}} \]

6. Applied egg-rr 19.3%

   \[\leadsto \color{blue}{\frac{\frac{-0.3333333333333333}{a} \cdot \left(b \cdot b - \mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)\right)}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}}} \]

7. Taylor expanded in a around 0

   \[\leadsto \frac{\color{blue}{-1 \cdot c}}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}} \]
8. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(c\right)}}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}} \]

   2. neg-lowering-neg.f64 99.9

      \[\leadsto \frac{\color{blue}{-c}}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}} \]

9. Simplified 99.9%

   \[\leadsto \frac{\color{blue}{-c}}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}} \]
10. Step-by-step derivation

    1. +-commutative N/A

       \[\leadsto \frac{\mathsf{neg}\left(c\right)}{b + \sqrt{\color{blue}{b \cdot b + c \cdot \left(a \cdot -3\right)}}} \]

    2. accelerator-lowering-fma.f64 N/A

       \[\leadsto \frac{\mathsf{neg}\left(c\right)}{b + \sqrt{\color{blue}{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)}}} \]

    3. *-lowering-*.f64 N/A

       \[\leadsto \frac{\mathsf{neg}\left(c\right)}{b + \sqrt{\mathsf{fma}\left(b, b, \color{blue}{c \cdot \left(a \cdot -3\right)}\right)}} \]

    4. *-lowering-*.f64 99.9

       \[\leadsto \frac{-c}{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \color{blue}{\left(a \cdot -3\right)}\right)}} \]

11. Applied egg-rr 99.9%

    \[\leadsto \frac{-c}{b + \sqrt{\color{blue}{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)}}} \]

12. Final simplification 99.9%

    \[\leadsto \frac{c}{\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)}} \]
13. Add Preprocessing

Alternative 2: 95.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \frac{-c}{\mathsf{fma}\left(a, \frac{c}{b} \cdot -1.5, b \cdot 2\right)} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (- c) (fma a (* (/ c b) -1.5) (* b 2.0))))

C

double code(double a, double b, double c) {
	return -c / fma(a, ((c / b) * -1.5), (b * 2.0));
}

Julia

function code(a, b, c)
	return Float64(Float64(-c) / fma(a, Float64(Float64(c / b) * -1.5), Float64(b * 2.0)))
end

Wolfram

code[a_, b_, c_] := N[((-c) / N[(a * N[(N[(c / b), $MachinePrecision] * -1.5), $MachinePrecision] + N[(b * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 19.0%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
2. Add Preprocessing

4. Applied egg-rr 19.0%

   \[\leadsto \color{blue}{\frac{-1}{\frac{3 \cdot a}{b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}}}} \]
5. Step-by-step derivation

   1. associate-/r/ N/A

      \[\leadsto \color{blue}{\frac{-1}{3 \cdot a} \cdot \left(b - \sqrt{a \cdot \left(-3 \cdot c\right) + b \cdot b}\right)} \]

   2. flip-- N/A

      \[\leadsto \frac{-1}{3 \cdot a} \cdot \color{blue}{\frac{b \cdot b - \sqrt{a \cdot \left(-3 \cdot c\right) + b \cdot b} \cdot \sqrt{a \cdot \left(-3 \cdot c\right) + b \cdot b}}{b + \sqrt{a \cdot \left(-3 \cdot c\right) + b \cdot b}}} \]

   3. associate-*r/ N/A

      \[\leadsto \color{blue}{\frac{\frac{-1}{3 \cdot a} \cdot \left(b \cdot b - \sqrt{a \cdot \left(-3 \cdot c\right) + b \cdot b} \cdot \sqrt{a \cdot \left(-3 \cdot c\right) + b \cdot b}\right)}{b + \sqrt{a \cdot \left(-3 \cdot c\right) + b \cdot b}}} \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{-1}{3 \cdot a} \cdot \left(b \cdot b - \sqrt{a \cdot \left(-3 \cdot c\right) + b \cdot b} \cdot \sqrt{a \cdot \left(-3 \cdot c\right) + b \cdot b}\right)}{b + \sqrt{a \cdot \left(-3 \cdot c\right) + b \cdot b}}} \]

6. Applied egg-rr 19.3%

   \[\leadsto \color{blue}{\frac{\frac{-0.3333333333333333}{a} \cdot \left(b \cdot b - \mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)\right)}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}}} \]

7. Taylor expanded in a around 0

   \[\leadsto \frac{\color{blue}{-1 \cdot c}}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}} \]
8. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(c\right)}}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}} \]

   2. neg-lowering-neg.f64 99.9

      \[\leadsto \frac{\color{blue}{-c}}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}} \]

9. Simplified 99.9%

   \[\leadsto \frac{\color{blue}{-c}}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}} \]

10. Taylor expanded in c around 0

    \[\leadsto \frac{\mathsf{neg}\left(c\right)}{\color{blue}{\frac{-3}{2} \cdot \frac{a \cdot c}{b} + 2 \cdot b}} \]
11. Step-by-step derivation

    1. *-commutative N/A

       \[\leadsto \frac{\mathsf{neg}\left(c\right)}{\color{blue}{\frac{a \cdot c}{b} \cdot \frac{-3}{2}} + 2 \cdot b} \]

    2. associate-/l* N/A

       \[\leadsto \frac{\mathsf{neg}\left(c\right)}{\color{blue}{\left(a \cdot \frac{c}{b}\right)} \cdot \frac{-3}{2} + 2 \cdot b} \]

    3. associate-*r* N/A

       \[\leadsto \frac{\mathsf{neg}\left(c\right)}{\color{blue}{a \cdot \left(\frac{c}{b} \cdot \frac{-3}{2}\right)} + 2 \cdot b} \]

    4. *-commutative N/A

       \[\leadsto \frac{\mathsf{neg}\left(c\right)}{a \cdot \color{blue}{\left(\frac{-3}{2} \cdot \frac{c}{b}\right)} + 2 \cdot b} \]

    5. accelerator-lowering-fma.f64 N/A

       \[\leadsto \frac{\mathsf{neg}\left(c\right)}{\color{blue}{\mathsf{fma}\left(a, \frac{-3}{2} \cdot \frac{c}{b}, 2 \cdot b\right)}} \]

    6. *-commutative N/A

       \[\leadsto \frac{\mathsf{neg}\left(c\right)}{\mathsf{fma}\left(a, \color{blue}{\frac{c}{b} \cdot \frac{-3}{2}}, 2 \cdot b\right)} \]

    7. *-lowering-*.f64 N/A

       \[\leadsto \frac{\mathsf{neg}\left(c\right)}{\mathsf{fma}\left(a, \color{blue}{\frac{c}{b} \cdot \frac{-3}{2}}, 2 \cdot b\right)} \]

    8. /-lowering-/.f64 N/A

       \[\leadsto \frac{\mathsf{neg}\left(c\right)}{\mathsf{fma}\left(a, \color{blue}{\frac{c}{b}} \cdot \frac{-3}{2}, 2 \cdot b\right)} \]

    9. *-commutative N/A

       \[\leadsto \frac{\mathsf{neg}\left(c\right)}{\mathsf{fma}\left(a, \frac{c}{b} \cdot \frac{-3}{2}, \color{blue}{b \cdot 2}\right)} \]

    10. *-lowering-*.f64 95.5

        \[\leadsto \frac{-c}{\mathsf{fma}\left(a, \frac{c}{b} \cdot -1.5, \color{blue}{b \cdot 2}\right)} \]

12. Simplified 95.5%

    \[\leadsto \frac{-c}{\color{blue}{\mathsf{fma}\left(a, \frac{c}{b} \cdot -1.5, b \cdot 2\right)}} \]
13. Add Preprocessing

Alternative 3: 95.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \frac{c}{\left(-b\right) - \mathsf{fma}\left(a, \frac{c}{b} \cdot -1.5, b\right)} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (/ c (- (- b) (fma a (* (/ c b) -1.5) b))))

C

double code(double a, double b, double c) {
	return c / (-b - fma(a, ((c / b) * -1.5), b));
}

Julia

function code(a, b, c)
	return Float64(c / Float64(Float64(-b) - fma(a, Float64(Float64(c / b) * -1.5), b)))
end

Wolfram

code[a_, b_, c_] := N[(c / N[((-b) - N[(a * N[(N[(c / b), $MachinePrecision] * -1.5), $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 19.0%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
2. Add Preprocessing

4. Applied egg-rr 19.0%

   \[\leadsto \color{blue}{\frac{-1}{\frac{3 \cdot a}{b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}}}} \]
5. Step-by-step derivation

   1. associate-/r/ N/A

      \[\leadsto \color{blue}{\frac{-1}{3 \cdot a} \cdot \left(b - \sqrt{a \cdot \left(-3 \cdot c\right) + b \cdot b}\right)} \]

   2. flip-- N/A

      \[\leadsto \frac{-1}{3 \cdot a} \cdot \color{blue}{\frac{b \cdot b - \sqrt{a \cdot \left(-3 \cdot c\right) + b \cdot b} \cdot \sqrt{a \cdot \left(-3 \cdot c\right) + b \cdot b}}{b + \sqrt{a \cdot \left(-3 \cdot c\right) + b \cdot b}}} \]

   3. associate-*r/ N/A

      \[\leadsto \color{blue}{\frac{\frac{-1}{3 \cdot a} \cdot \left(b \cdot b - \sqrt{a \cdot \left(-3 \cdot c\right) + b \cdot b} \cdot \sqrt{a \cdot \left(-3 \cdot c\right) + b \cdot b}\right)}{b + \sqrt{a \cdot \left(-3 \cdot c\right) + b \cdot b}}} \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{-1}{3 \cdot a} \cdot \left(b \cdot b - \sqrt{a \cdot \left(-3 \cdot c\right) + b \cdot b} \cdot \sqrt{a \cdot \left(-3 \cdot c\right) + b \cdot b}\right)}{b + \sqrt{a \cdot \left(-3 \cdot c\right) + b \cdot b}}} \]

6. Applied egg-rr 19.3%

   \[\leadsto \color{blue}{\frac{\frac{-0.3333333333333333}{a} \cdot \left(b \cdot b - \mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)\right)}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}}} \]

7. Taylor expanded in a around 0

   \[\leadsto \frac{\color{blue}{-1 \cdot c}}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}} \]
8. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(c\right)}}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}} \]

   2. neg-lowering-neg.f64 99.9

      \[\leadsto \frac{\color{blue}{-c}}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}} \]

9. Simplified 99.9%

   \[\leadsto \frac{\color{blue}{-c}}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}} \]

10. Taylor expanded in c around 0

    \[\leadsto \frac{\mathsf{neg}\left(c\right)}{b + \color{blue}{\left(b + \frac{-3}{2} \cdot \frac{a \cdot c}{b}\right)}} \]
11. Step-by-step derivation

    1. +-commutative N/A

       \[\leadsto \frac{\mathsf{neg}\left(c\right)}{b + \color{blue}{\left(\frac{-3}{2} \cdot \frac{a \cdot c}{b} + b\right)}} \]

    2. *-commutative N/A

       \[\leadsto \frac{\mathsf{neg}\left(c\right)}{b + \left(\color{blue}{\frac{a \cdot c}{b} \cdot \frac{-3}{2}} + b\right)} \]

    3. associate-/l* N/A

       \[\leadsto \frac{\mathsf{neg}\left(c\right)}{b + \left(\color{blue}{\left(a \cdot \frac{c}{b}\right)} \cdot \frac{-3}{2} + b\right)} \]

    4. associate-*r* N/A

       \[\leadsto \frac{\mathsf{neg}\left(c\right)}{b + \left(\color{blue}{a \cdot \left(\frac{c}{b} \cdot \frac{-3}{2}\right)} + b\right)} \]

    5. *-commutative N/A

       \[\leadsto \frac{\mathsf{neg}\left(c\right)}{b + \left(a \cdot \color{blue}{\left(\frac{-3}{2} \cdot \frac{c}{b}\right)} + b\right)} \]

    6. accelerator-lowering-fma.f64 N/A

       \[\leadsto \frac{\mathsf{neg}\left(c\right)}{b + \color{blue}{\mathsf{fma}\left(a, \frac{-3}{2} \cdot \frac{c}{b}, b\right)}} \]

    7. *-commutative N/A

       \[\leadsto \frac{\mathsf{neg}\left(c\right)}{b + \mathsf{fma}\left(a, \color{blue}{\frac{c}{b} \cdot \frac{-3}{2}}, b\right)} \]

    8. *-lowering-*.f64 N/A

       \[\leadsto \frac{\mathsf{neg}\left(c\right)}{b + \mathsf{fma}\left(a, \color{blue}{\frac{c}{b} \cdot \frac{-3}{2}}, b\right)} \]

    9. /-lowering-/.f64 95.5

       \[\leadsto \frac{-c}{b + \mathsf{fma}\left(a, \color{blue}{\frac{c}{b}} \cdot -1.5, b\right)} \]

12. Simplified 95.5%

    \[\leadsto \frac{-c}{b + \color{blue}{\mathsf{fma}\left(a, \frac{c}{b} \cdot -1.5, b\right)}} \]

13. Final simplification 95.5%

    \[\leadsto \frac{c}{\left(-b\right) - \mathsf{fma}\left(a, \frac{c}{b} \cdot -1.5, b\right)} \]
14. Add Preprocessing

Alternative 4: 90.3% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \frac{c}{b} \cdot -0.5 \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (* (/ c b) -0.5))

C

double code(double a, double b, double c) {
	return (c / b) * -0.5;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (c / b) * (-0.5d0)
end function

Java

public static double code(double a, double b, double c) {
	return (c / b) * -0.5;
}

Python

def code(a, b, c):
	return (c / b) * -0.5

Julia

function code(a, b, c)
	return Float64(Float64(c / b) * -0.5)
end

MATLAB

function tmp = code(a, b, c)
	tmp = (c / b) * -0.5;
end

Wolfram

code[a_, b_, c_] := N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]

Derivation

1. Initial program 19.0%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
2. Add Preprocessing

3. Taylor expanded in b around inf

   \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b}} \]
4. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b}} \]

   2. /-lowering-/.f64 89.5

      \[\leadsto -0.5 \cdot \color{blue}{\frac{c}{b}} \]

5. Simplified 89.5%

   \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b}} \]

6. Final simplification 89.5%

   \[\leadsto \frac{c}{b} \cdot -0.5 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024204 
(FPCore (a b c)
  :name "Cubic critical, wide range"
  :precision binary64
  :pre (and (and (and (< 4.930380657631324e-32 a) (< a 2.028240960365167e+31)) (and (< 4.930380657631324e-32 b) (< b 2.028240960365167e+31))) (and (< 4.930380657631324e-32 c) (< c 2.028240960365167e+31)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))